Show that the language

$L=\{a^n{b}^k{c}^n$:$n0,kn\}$

is not regular.

$3.$ Show that the language

$L=\{a^n{b}^k{c}^n{d}^k$:$n0,k>n\}$

is not regular.

$4.$ Show that the language

$L=\{w$:$n_a(w)=n_b(w)\}$

is not regular. Is $L^{*}$ regular?

$5.$ Prove that the following languages are not regular:

$($a$)L=\{a^n{b}^l{a}^k$:$kn+l\}.$

$($b$)L=\{a^n{b}^l{a}^k$:$kn+l\}.$

$($c$)L=\{\begin{array}{c}{a}^n{b}^l{a}^k\text{:}n=l\end{array}$ or $\{$:$lk\}.$

$($d$)L=\{a^n{b}^l$:$nl\}.$

$($e$)L=\{w$:$n_a(w)n_b(w)\}.$

$($f$)L=\{ww$:win$\{a,b{\}}^{*}\}.$

$($g$)L=\{w^{R}ww{w}^R$:win$\{a,b{\}}^{*}\}.$

$6.$ Determine whether or not the following languages on $=\{a\}$ are regular:

$($a$)L=\{\begin{array}{c}{a}^n\text{:}n2\end{array}$ is a prime number